Then \((c_i)\)  is a Thue-Morse type sequence generated by \(c_0^-\)  (cf. [1]}).
From [2]} it follows that for each \(\ell =1,2,\cdots \)  there exists \(q_\ell \in (q_0, q_0^*)\)  such that \(\beta (q_\ell )=c_\ell 0^\infty \) . de Vries and Komornik
obtained in [3]}, [2]} that for each connected component \((q_0,q^*_0)\)
\(\mathcal {V}\cap (q_0,q^*_0)=\left\lbrace q_\ell ;\ell \in \mathbb {N}\right\rbrace \;\textrm {and}\; q_\ell \uparrow q^*_0.\) 
